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Journal Cover Physica D: Nonlinear Phenomena
  [SJR: 1.049]   [H-I: 102]   [3 followers]  Follow
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
   ISSN (Print) 0167-2789
   Published by Elsevier Homepage  [3177 journals]
  • Multidimensional equilibria and their stability in copolymer–solvent
    • Authors: Karl Glasner; Saulo Orizaga
      Pages: 1 - 12
      Abstract: Publication date: 15 June 2018
      Source:Physica D: Nonlinear Phenomena, Volume 373
      Author(s): Karl Glasner, Saulo Orizaga
      This paper discusses localized equilibria which arise in copolymer–solvent mixtures. A free boundary problem associated with the sharp-interface limit of a density functional model is used to identify both lamellar and concentric domain patterns composed of a finite number of layers. Stability of these morphologies is studied through explicit linearization of the free boundary evolution. For the multilayered lamellar configuration, transverse instability is observed for sufficiently small dimensionless interfacial energies. Additionally, a crossover between small and large wavelength instabilities is observed depending on whether solvent–polymer or monomer–monomer interfacial energy is dominant. Concentric domain patterns resembling multilayered micelles and vesicles exhibit bifurcations wherein they only exist for sufficiently small dimensionless interfacial energies. The bifurcation of large radii vesicle solutions is studied analytically, and a crossover from a supercritical case with only one solution branch to a subcritical case with two is observed. Linearized stability of these configurations shows that azimuthal perturbation may lead to instabilities as interfacial energy is decreased.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.02.001
      Issue No: Vol. 373 (2018)
  • Geometric chaos indicators and computations of the spherical hypertube
           manifolds of the spatial circular restricted three-body problem
    • Authors: Massimiliano Guzzo; Elena Lega
      Pages: 38 - 58
      Abstract: Publication date: 15 June 2018
      Source:Physica D: Nonlinear Phenomena, Volume 373
      Author(s): Massimiliano Guzzo, Elena Lega
      The circular restricted three-body problem has five relative equilibria L 1 , L 2 , . . . , L 5 . The invariant stable–unstable manifolds of the center manifolds originating at the partially hyperbolic equilibria L 1 , L 2 have been identified as the separatrices for the motions which transit between the regions of the phase-space which are internal or external with respect to the two massive bodies. While the stable and unstable manifolds of the planar problem have been extensively studied both theoretically and numerically, the spatial case has not been as deeply investigated. This paper is devoted to the global computation of these manifolds in the spatial case with a suitable finite time chaos indicator. The definition of the chaos indicator is not trivial, since the mandatory use of the regularizing Kustaanheimo–Stiefel variables may introduce discontinuities in the finite time chaos indicators. From the study of such discontinuities, we define geometric chaos indicators which are globally defined and smooth, and whose ridges sharply approximate the stable and unstable manifolds of the center manifolds of L 1 , L 2 . We illustrate the method for the Sun–Jupiter mass ratio, and represent the topology of the asymptotic manifolds using sections and three-dimensional representations.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.02.003
      Issue No: Vol. 373 (2018)
  • On the nonexistence of degenerate phase-shift discrete solitons in a dNLS
           nonlocal lattice
    • Authors: T. Penati; M. Sansottera; S. Paleari; V. Koukouloyannis; P.G. Kevrekidis
      Pages: 1 - 13
      Abstract: Publication date: 1 May 2018
      Source:Physica D: Nonlinear Phenomena, Volume 370
      Author(s): T. Penati, M. Sansottera, S. Paleari, V. Koukouloyannis, P.G. Kevrekidis
      We consider a one-dimensional discrete nonlinear Schrödinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to four-site vortex solutions in the standard two-dimensional dNLS model (square lattice), of which this is a simpler variant. Due to the specific choice of lengths of the inter-site interactions, the vortex configurations considered present a degeneracy which causes the standard continuation techniques to be non-applicable. In the present one-dimensional case, the existence of a conserved quantity for the soliton profile (the so-called density current), together with a perturbative construction, leads to the nonexistence of any phase-shift discrete soliton which is at least C 2 with respect to the small coupling ϵ , in the limit of vanishing ϵ . If we assume the solution to be only C 0 in the same limit of ϵ , nonexistence is instead proved by studying the bifurcation equation of a Lyapunov–Schmidt reduction, expanded to suitably high orders. Specifically, we produce a nonexistence criterion whose efficiency we reveal in the cases of partial and full degeneracy of approximate solutions obtained via a leading order expansion.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2017.12.012
      Issue No: Vol. 370 (2018)
  • On the classification of the spectrally stable standing waves of the
           Hartree problem
    • Authors: Vladimir Georgiev; Atanas Stefanov
      Pages: 29 - 39
      Abstract: Publication date: 1 May 2018
      Source:Physica D: Nonlinear Phenomena, Volume 370
      Author(s): Vladimir Georgiev, Atanas Stefanov
      We consider the fractional Hartree model, with general power non-linearity and arbitrary spatial dimension. We construct variationally the “normalized” solutions for the corresponding Choquard–Pekar model—in particular a number of key properties, like smoothness and bell-shapedness are established. As a consequence of the construction, we show that these solitons are spectrally stable as solutions to the time-dependent Hartree model. In addition, we analyze the spectral stability of the Moroz–Van Schaftingen solitons of the classical Hartree problem, in any dimensions and power non-linearity. A full classification is obtained, the main conclusion of which is that only and exactly the “normalized” solutions (which exist only in a portion of the range) are spectrally stable.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.01.002
      Issue No: Vol. 370 (2018)
  • Concentration and limit behaviors of stationary measures
    • Authors: Wen Huang; Min Ji; Zhenxin Liu; Yingfei Yi
      Pages: 1 - 17
      Abstract: Publication date: 15 April 2018
      Source:Physica D: Nonlinear Phenomena, Volume 369
      Author(s): Wen Huang, Min Ji, Zhenxin Liu, Yingfei Yi
      In this paper, we study limit behaviors of stationary measures of the Fokker–Planck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative noise including additive white noise case. As the noises are vanishing, various results on the invariance and concentration of the limit measures are obtained. In particular, we show that if the noise perturbed systems admit a uniform Lyapunov function, then the stationary measures form a relatively sequentially compact set whose weak ∗ -limits are invariant measures of the unperturbed system concentrated on its global attractor. In the case that the global attractor contains a strong local attractor, we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor; and on the contrary, in the presence of a strong local repeller in the global attractor, there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller. Moreover, we show that if there is a strongly repelling equilibrium in the global attractor, then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium. As applications of these results, an example of stochastic Hopf bifurcation and an example with non-decomposable ω -limit sets are provided. Our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2017.12.009
      Issue No: Vol. 369 (2018)
  • Stabilizing the long-time behavior of the forced Navier–Stokes and
           damped Euler systems by large mean flow
    • Authors: Jacek Cyranka; Piotr B. Mucha; Edriss S. Titi; Piotr Zgliczyński
      Pages: 18 - 29
      Abstract: Publication date: 15 April 2018
      Source:Physica D: Nonlinear Phenomena, Volume 369
      Author(s): Jacek Cyranka, Piotr B. Mucha, Edriss S. Titi, Piotr Zgliczyński
      The paper studies the issue of stability of solutions to the forced Navier–Stokes and damped Euler systems in periodic boxes. It is shown that for large, but fixed, Grashoff (Reynolds) number the turbulent behavior of all Leray–Hopf weak solutions of the three-dimensional Navier–Stokes equations, in periodic box, is suppressed, when viewed in the right frame of reference, by large enough average flow of the initial data; a phenomenon that is similar in spirit to the Landau damping. Specifically, we consider an initial data which have large enough spatial average, then by means of the Galilean transformation, and thanks to the periodic boundary conditions, the large time independent forcing term changes into a highly oscillatory force; which then allows us to employ some averaging principles to establish our result. Moreover, we also show that under the action of fast oscillatory-in-time external forces all two-dimensional regular solutions of the Navier–Stokes and the damped Euler equations converge to a unique time-periodic solution.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2017.12.010
      Issue No: Vol. 369 (2018)
  • On common noise-induced synchronization in complex networks with
           state-dependent noise diffusion processes
    • Authors: Giovanni Russo; Robert Shorten
      Pages: 47 - 54
      Abstract: Publication date: 15 April 2018
      Source:Physica D: Nonlinear Phenomena, Volume 369
      Author(s): Giovanni Russo, Robert Shorten
      This paper is concerned with the study of common noise-induced synchronization phenomena in complex networks of diffusively coupled nonlinear systems. We consider the case where common noise propagation depends on the network state and, as a result, the noise diffusion process at the nodes depends on the state of the network. For such networks, we present an algebraic sufficient condition for the onset of synchronization, which depends on the network topology, the dynamics at the nodes, the coupling strength and the noise diffusion. Our result explicitly shows that certain noise diffusion processes can drive an unsynchronized network towards synchronization. In order to illustrate the effectiveness of our result, we consider two applications: collective decision processes and synchronization of chaotic systems. We explicitly show that, in the former application, a sufficiently large noise can drive a population towards a common decision, while, in the latter, we show how common noise can synchronize a network of Lorentz chaotic systems.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.01.003
      Issue No: Vol. 369 (2018)
  • Gradual multifractal reconstruction of time-series: Formulation of the
           method and an application to the coupling between stock market indices and
           their Hölder exponents
    • Authors: Christopher J. Keylock
      Pages: 1 - 9
      Abstract: Publication date: 1 April 2018
      Source:Physica D: Nonlinear Phenomena, Volume 368
      Author(s): Christopher J. Keylock
      A technique termed gradual multifractal reconstruction (GMR) is formulated. A continuum is defined from a signal that preserves the pointwise Hölder exponent (multifractal) structure of a signal but randomises the locations of the original data values with respect to this ( φ = 0 ), to the original signal itself( φ = 1 ). We demonstrate that this continuum may be populated with synthetic time series by undertaking selective randomisation of wavelet phases using a dual-tree complex wavelet transform. That is, the φ = 0 end of the continuum is realised using the recently proposed iterated, amplitude adjusted wavelet transform algorithm (Keylock, 2017) that fully randomises the wavelet phases. This is extended to the GMR formulation by selective phase randomisation depending on whether or not the wavelet coefficient amplitudes exceeds a threshold criterion. An econophysics application of the technique is presented. The relation between the normalised log-returns and their Hölder exponents for the daily returns of eight financial indices are compared. One particularly noticeable result is the change for the two American indices (NASDAQ 100 and S&P 500) from a non-significant to a strongly significant (as determined using GMR) cross-correlation between the returns and their Hölder exponents from before the 2008 crash to afterwards. This is also reflected in the skewness of the phase difference distributions, which exhibit a geographical structure, with Asian markets not exhibiting significant skewness in contrast to those from elsewhere globally.

      PubDate: 2018-02-26T14:48:37Z
      DOI: 10.1016/j.physd.2017.11.011
      Issue No: Vol. 368 (2018)
  • Synchronisation under shocks: The Lévy Kuramoto model
    • Authors: Dale Roberts; Alexander C. Kalloniatis
      Pages: 10 - 21
      Abstract: Publication date: 1 April 2018
      Source:Physica D: Nonlinear Phenomena, Volume 368
      Author(s): Dale Roberts, Alexander C. Kalloniatis
      We study the Kuramoto model of identical oscillators on Erdős–Rényi (ER) and Barabasi–Alberts (BA) scale free networks examining the dynamics when perturbed by a Lévy noise. Lévy noise exhibits heavier tails than Gaussian while allowing for their tempering in a controlled manner. This allows us to understand how ‘shocks’ influence individual oscillator and collective system behaviour of a paradigmatic complex system. Skewed α -stable Lévy noise, equivalent to fractional diffusion perturbations, are considered, but overlaid by exponential tempering of rate λ . In an earlier paper we found that synchrony takes a variety of forms for identical Kuramoto oscillators subject to stable Lévy noise, not seen for the Gaussian case, and changing with α : a noise-induced drift, a smooth α dependence of the point of cross-over of synchronisation point of ER and BA networks, and a severe loss of synchronisation at low values of α . In the presence of tempering we observe both analytically and numerically a dramatic change to the α < 1 behaviour where synchronisation is sustained over a larger range of values of the ‘noise strength’ σ , improved compared to the α > 1 tempered cases. Analytically we study the system close to the phase synchronised fixed point and solve the tempered fractional Fokker–Planck equation. There we observe that densities show stronger support in the basin of attraction at low α for fixed coupling, σ and tempering λ . We then perform numerical simulations for networks of size N = 1000 and average degree d ̄ = 10 . There, we compute the order parameter r as a function of σ for fixed α and λ and observe values of r ≈ 1 over larger ranges of σ for α < 1 and λ ≠ 0 . In addition we observe drift of both positive and negative slopes for different α and λ when native frequencies are equal, and confirm a sustainment of synchronisation down to low values of α . We propose a mechanism for this in terms of the basic shape of the tempered stable Lévy densities for various α and how it feeds into Kuramoto oscillator dynamics and illustrate this with examples of specific paths.

      PubDate: 2018-02-26T14:48:37Z
      DOI: 10.1016/j.physd.2017.12.005
      Issue No: Vol. 368 (2018)
  • Simultaneous estimation of deterministic and fractal stochastic components
           in non-stationary time series
    • Authors: Constantino A. García; Abraham Otero; Paulo Félix; Jesús Presedo; David G. Márquez
      Abstract: Publication date: Available online 24 April 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Constantino A. García, Abraham Otero, Paulo Félix, Jesús Presedo, David G. Márquez
      In the past few decades, it has been recognized that 1 ∕ f fluctuations are ubiquitous in nature. The most widely used mathematical models to capture the long-term memory properties of 1 ∕ f fluctuations have been stochastic fractal models. However, physical systems do not usually consist on just stochastic fractal dynamics, but they often also show some degree of deterministic behavior. The present paper proposes a model based on fractal stochastic and deterministic components that can provide a valuable basis for the study of complex systems with long-term correlations. The fractal stochastic component is assumed to be a fractional Brownian motion process and the deterministic component is assumed to be a band-limited signal. We also provide a method that, under the assumptions of this model, is able to characterize the fractal stochastic component and to provide an estimate of the deterministic components present in a given time series. The method is based on a Bayesian wavelet shrinkage procedure that exploits the self-similar properties of the fractal processes in the wavelet domain. This method has been validated over simulated signals and over real signals with economical and biological origin. Real examples illustrate how our model may be useful for exploring the deterministic-stochastic duality of complex systems, and uncovering interesting patterns present in time series.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.04.002
  • Entire solutions originating from monotone fronts to the Allen-Cahn
    • Authors: Yan-Yu Chen; Jong-Shenq Guo; Nirokazu Ninomiya; Chih-Hong Yao
      Abstract: Publication date: Available online 19 April 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Yan-Yu Chen, Jong-Shenq Guo, Nirokazu Ninomiya, Chih-Hong Yao
      In this paper, we study entire solutions of the Allen-Cahn equation in one-dimensional Euclidean space. This equation is a scalar reaction–diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions originating from three and four fronts is shown by constructing some suitable pairs of super-sub-solutions. Moreover, we show that there are no entire solutions originating from more than four fronts.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.04.003
  • Visibility graphs and symbolic dynamics
    • Authors: Lucas Lacasa; Wolfram Just
      Abstract: Publication date: Available online 11 April 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Lucas Lacasa, Wolfram Just
      Visibility algorithms are a family of geometric and ordering criteria by which a real-valued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology nontrivial properties of the series structure, and can thus be seen as a combinatorial representation of a dynamical system. Here we explore in some detail the relation between visibility graphs and symbolic dynamics. To do that, we consider the degree sequence of horizontal visibility graphs generated by the one-parameter logistic map, for a range of values of the parameter for which the map shows chaotic behaviour. Numerically, we observe that in the chaotic region the block entropies of these sequences systematically converge to the Lyapunov exponent of the time series. Hence, Pesin’s identity suggests that these block entropies are converging to the Kolmogorov–Sinai entropy of the physical measure, which ultimately suggests that the algorithm is implicitly and adaptively constructing phase space partitions which might have the generating property. To give analytical insight, we explore the relation k ( x ) , x ∈ [ 0 , 1 ] that, for a given datum with value x , assigns in graph space a node with degree k . In the case of the out-degree sequence, such relation is indeed a piece-wise constant function. By making use of explicit methods and tools from symbolic dynamics we are able to analytically show that the algorithm indeed performs an effective partition of the phase space and that such partition is naturally expressed as a countable union of subintervals, where the endpoints of each subinterval are related to the fixed point structure of the iterates of the map and the subinterval enumeration is associated with particular ordering structures that we called motifs.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.04.001
  • Determining modes for the surface quasi-geostrophic equation
    • Authors: Alexey Cheskidov; Mimi Dai
      Abstract: Publication date: Available online 27 March 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexey Cheskidov, Mimi Dai
      We introduce a determining wavenumber for the surface quasi-geostrophic (SQG) equation defined for each individual trajectory and then study its dependence on the force. While in the subcritical and critical cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, and measures the number of degrees of freedom of the flow, or resolution needed to describe a solution to the SQG equation.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.03.003
  • On 3D Navier–Stokes equations: Regularization and uniqueness by
    • Authors: Hakima Bessaih; María J. Garrido-Atienza; Björn Schmalfuß
      Abstract: Publication date: Available online 27 March 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Hakima Bessaih, María J. Garrido-Atienza, Björn Schmalfuß
      A modified version of the three dimensional Navier–Stokes equations is considered with periodic boundary conditions. A bounded constant delay is introduced into the convective term, that produces a regularizing effect on the solution. In fact, by assuming appropriate regularity on the initial data, the solutions of the delayed equations are proved to be regular and, as a consequence, existence and also uniqueness of a global weak solution is obtained. Moreover, the associated flow is constructed and the continuity of the semigroup is proved. Finally, we investigate the passage to the limit on the delay, obtaining that the limit is a weak solution of the Navier–Stokes equations.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.03.004
  • Pathwise upper semi-continuity of random pullback attractors along the
           time axis
    • Authors: Hongyong Cui; Peter E. Kloeden; Fuke Wu
      Abstract: Publication date: Available online 20 March 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Hongyong Cui, Peter E. Kloeden, Fuke Wu
      The pullback attractor of a non-autonomous random dynamical system is a time-indexed family of random sets, typically having the form { A t ( ⋅ ) } t ∈ R with each A t ( ⋅ ) a random set. This paper is concerned with the nature of such time-dependence. It is shown that the upper semi-continuity of the mapping t ↦ A t ( ω ) for each ω fixed has an equivalence relationship with the uniform compactness of the local union ∪ s ∈ I A s ( ω ) , where I ⊂ R is compact. Applied to a semi-linear degenerate parabolic equation with additive noise and a wave equation with multiplicative noise we show that, in order to prove the above locally uniform compactness and upper semi-continuity, no additional conditions are required, in which sense the two properties appear to be general properties satisfied by a large number of real models.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.03.002
  • Dynamics of a linear system coupled to a chain of light nonlinear
           oscillators analyzed through a continuous approximation
    • Authors: S. Charlemagne; A. Ture Savadkoohi; C.-H. Lamarque
      Abstract: Publication date: Available online 13 March 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): S. Charlemagne, A. Ture Savadkoohi, C.-H. Lamarque
      The continuous approximation is used in this work to describe the dynamics of a nonlinear chain of light oscillators coupled to a linear main system. A general methodology is applied to an example where the chain has local nonlinear restoring forces. The slow invariant manifold is detected at fast time scale. At slow time scale, equilibrium and singular points are sought around this manifold in order to predict periodic regimes and strongly modulated responses of the system. Analytical predictions are in good accordance with numerical results and represent a potent tool for designing nonlinear chains for passive control purposes.

      PubDate: 2018-04-25T08:59:04Z
      DOI: 10.1016/j.physd.2018.03.001
  • Turbulence in vertically averaged convection
    • Authors: N. Balci; A.M. Isenberg; M.S. Jolly
      Abstract: Publication date: Available online 23 February 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): N. Balci, A.M. Isenberg, M.S. Jolly
      The vertically averaged velocity of the 3D Rayleigh-Bénard problem is analyzed and numerically simulated. This vertically averaged velocity satisfies a 2D incompressible Navier–Stokes system with a body force involving the 3D velocity. A time average of this force is estimated through time averages of the 3D velocity. Relations similar to those from 2D turbulence are then derived. Direct numerical simulation of the 3D Rayleigh Bénard are carried out to test how prominent the features of 2D turbulence are for this Navier–Stokes system.

      PubDate: 2018-02-26T14:48:37Z
      DOI: 10.1016/j.physd.2018.02.005
  • Wave breaking for the Stochastic Camassa–Holm equation
    • Authors: Dan Crisan; Darryl D. Holm
      Abstract: Publication date: Available online 16 February 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Dan Crisan, Darryl D. Holm
      We show that wave breaking occurs with positive probability for the Stochastic Camassa–Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the time-asymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space–time paths.

      PubDate: 2018-02-26T14:48:37Z
      DOI: 10.1016/j.physd.2018.02.004
  • The stability and slow dynamics of spot patterns in the 2D Brusselator
           model: The effect of open systems and heterogeneities
    • Authors: J.C. Tzou; M.J. Ward
      Abstract: Publication date: Available online 13 February 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): J.C. Tzou, M.J. Ward
      Spot patterns, whereby the activator field becomes spatially localized near certain dynamically-evolving discrete spatial locations in a bounded multi-dimensional domain, is a common occurrence for two-component reaction–diffusion (RD) systems in the singular limit of a large diffusivity ratio. In previous studies of 2-D localized spot patterns for various specific well-known RD systems, the domain boundary was assumed to be impermeable to both the activator and inhibitor, and the reaction-kinetics were assumed to be spatially uniform. As an extension of this previous theory, we use formal asymptotic methods to study the existence, stability, and slow dynamics of localized spot patterns for the singularly perturbed 2-D Brusselator RD model when the domain boundary is only partially impermeable, as modeled by an inhomogeneous Robin boundary condition, or when there is an influx of inhibitor across the domain boundary. In our analysis, we will also allow for the effect of a spatially variable bulk feed term in the reaction kinetics. By applying our extended theory to the special case of one-spot patterns and ring patterns of spots inside the unit disk, we provide a detailed analysis of the effect on spot patterns of these three different sources of heterogeneity. In particular, when there is an influx of inhibitor across the boundary of the unit disk, a ring pattern of spots can become pinned to a ring-radius closer to the domain boundary. Under a Robin condition, a quasi-equilibrium ring pattern of spots is shown to exhibit a novel saddle–node bifurcation behavior in terms of either the inhibitor diffusivity, the Robin constant, or the ambient background concentration. A spatially variable bulk feed term, with a concentrated source of “fuel” inside the domain, is shown to yield a saddle–node bifurcation structure of spot equilibria, which leads to qualitatively new spot-pinning behavior. Results from our asymptotic theory are validated from full numerical simulations of the Brusselator model.

      PubDate: 2018-02-26T14:48:37Z
      DOI: 10.1016/j.physd.2018.02.002
  • Krein signature for instability of PT-symmetric states
    • Authors: Alexander Chernyavsky; Dmitry E. Pelinovsky
      Abstract: Publication date: Available online 9 February 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexander Chernyavsky, Dmitry E. Pelinovsky
      Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the PT -symmetric nonlinear Schrödinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature.

      PubDate: 2018-02-26T14:48:37Z
      DOI: 10.1016/j.physd.2018.01.009
  • Scale-free behavior of networks with the copresence of preferential and
           uniform attachment rules
    • Authors: Angelica Pachon; Laura Sacerdote; Shuyi Yang
      Abstract: Publication date: Available online 3 February 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Angelica Pachon, Laura Sacerdote, Shuyi Yang
      Complex networks in different areas exhibit degree distributions with a heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scale-free behavior of the degree distribution. We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1 − p ) that stresses the rich get richer system, and a uniform choice (with probability p ) for the most recent nodes, i.e. the nodes belonging to a window of size w to the left of the last born node. The latter highlights a trend to select one of the last added nodes when no information is available. The recent nodes can be either a given fixed number or a proportion ( α n ) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically power-law degree distribution. The same result is then illustrated through simulations in the second case. When the window of recent nodes has a constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold. The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2018.01.005
  • A third-order class-D amplifier with and without ripple compensation
    • Authors: Stephen M. Cox; H. du Toit Mouton
      Abstract: Publication date: Available online 3 February 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Stephen M. Cox, H. du Toit Mouton
      We analyse the nonlinear behaviour of a third-order class-D amplifier, and demonstrate the remarkable effectiveness of the recently introduced ripple compensation (RC) technique in reducing the audio distortion of the device. The amplifier converts an input audio signal to a high-frequency train of rectangular pulses, whose widths are modulated according to the input signal (pulse-width modulation) and employs negative feedback. After determining the steady-state operating point for constant input and calculating its stability, we derive a small-signal model (SSM), which yields in closed form the transfer function relating (infinitesimal) input and output disturbances. This SSM shows how the RC technique is able to linearise the small-signal response of the device. We extend this SSM through a fully nonlinear perturbation calculation of the dynamics of the amplifier, based on the disparity in time scales between the pulse train and the audio signal. We obtain the nonlinear response of the amplifier to a general audio signal, avoiding the linearisation inherent in the SSM; we thereby more precisely quantify the reduction in distortion achieved through RC. Finally, simulations corroborate our theoretical predictions and illustrate the dramatic deterioration in performance that occurs when the amplifier is operated in an unstable regime. The perturbation calculation is rather general, and may be adapted to quantify the way in which other nonlinear negative-feedback pulse-modulated devices track a time-varying input signal that slowly modulates the system parameters.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2018.01.012
  • Pseudo-simple heteroclinic cycles in R4
    • Authors: Pascal Chossat; Alexander Lohse; Olga Podvigina
      Abstract: Publication date: Available online 2 February 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Pascal Chossat, Alexander Lohse, Olga Podvigina
      We study pseudo-simple heteroclinic cycles for a Γ -equivariant system in R 4 with finite Γ ⊂ O ( 4 ) , and their nearby dynamics. In particular, in a first step towards a full classification –analogous to that which exists already for the class of simple cycles –we identify all finite subgroups of O ( 4 ) admitting pseudo-simple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudo-simple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups Γ ⊂ O ( 4 ) , Γ ⊄ S O ( 4 ) , admitting fragmentarily asymptotically stable pseudo-simple heteroclinic cycles. (It has been previously shown that for Γ ⊂ S O ( 4 ) pseudo-simple cycles generically are completely unstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudo-simple cycle as a subset.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2018.01.008
  • Exact closed-form solutions of a fully nonlinear asymptotic two-fluid
    • Authors: Alexei F. Cheviakov
      Abstract: Publication date: Available online 31 January 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexei F. Cheviakov
      A fully nonlinear model of Choi and Camassa (1999) describing one-dimensional incompressible dynamics of two non-mixing fluids in a horizontal channel, under a shallow water approximation, is considered. An equivalence transformation is presented, leading to a special dimensionless form of the system, involving a single dimensionless constant physical parameter, as opposed to five parameters present in the original model. A first-order dimensionless ordinary differential equation describing traveling wave solutions is analyzed. Several multi-parameter families of physically meaningful exact closed-form solutions of the two-fluid model are derived, corresponding to periodic, solitary, and kink-type bidirectional traveling waves; specific examples are given, and properties of the exact solutions are analyzed.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2018.01.001
  • Limit cycles via higher order perturbations for some piecewise
           differential systems
    • Authors: Claudio A. Buzzi; Maurício Firmino Silva Lima; Joan Torregrosa
      Abstract: Publication date: Available online 31 January 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Claudio A. Buzzi, Maurício Firmino Silva Lima, Joan Torregrosa
      A classical perturbation problem is the polynomial perturbation of the harmonic oscillator, ( x ′ , y ′ ) = ( − y + ε f ( x , y , ε ) , x + ε g ( x , y , ε ) ) . In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n , no more than N n − 1 limit cycles appear up to a study of order N . We also show that this upper bound is reached for orders one and two. Moreover, we study this problem in some classes of piecewise Liénard differential systems providing better upper bounds for higher order perturbation in ε , showing also when they are reached. The Poincaré–Pontryagin–Melnikov theory is the main technique used to prove all the results.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2018.01.007
  • Bifurcation analysis of eight coupled degenerate optical parametric
    • Authors: Daisuke Ito; Tetsushi Ueta; Kazuyuki Aihara
      Abstract: Publication date: Available online 31 January 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Daisuke Ito, Tetsushi Ueta, Kazuyuki Aihara
      A degenerate optical parametric oscillator (DOPO) network realized as a coherent Ising machine can be used to solve combinatorial optimization problems. Both theoretical and experimental investigations into the performance of DOPO networks have been presented previously. However a problem remains, namely that the dynamics of the DOPO network itself can lower the search success rates of globally optimal solutions for Ising problems. This paper shows that the problem is caused by pitchfork bifurcations due to the symmetry structure of coupled DOPOs. Some two-parameter bifurcation diagrams of equilibrium points express the performance deterioration. It is shown that the emergence of non-ground states regarding local minima hampers the system from reaching the ground states corresponding to the global minimum. We then describe a parametric strategy for leading a system to the ground state by actively utilizing the bifurcation phenomena. By adjusting the parameters to break particular symmetry, we find appropriate parameter sets that allow the coherent Ising machine to obtain the globally optimal solution alone.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2018.01.010
  • Generalized Lagrangian coherent structures
    • Authors: Sanjeeva Balasuriya; Nicholas T. Ouellette; Irina I. Rypina
      Abstract: Publication date: Available online 31 January 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Sanjeeva Balasuriya, Nicholas T. Ouellette, Irina I. Rypina
      The notion of a Lagrangian Coherent Structure (LCS) is by now well established as a way to capture transient coherent transport dynamics in unsteady and aperiodic fluid flows that are known over finite time. We show that the concept of an LCS can be generalized to capture coherence in other quantities of interest that are transported by, but not fully locked to, the fluid. Such quantities include those with dynamic, biological, chemical, or thermodynamic relevance, such as temperature, pollutant concentration, vorticity, kinetic energy, plankton density, and so on. We provide a conceptual framework for identifying the Generalized Lagrangian Coherent Structures (GLCSs) associated with such evolving quantities. We show how LCSs can be seen as a special case within this framework, and provide an overarching discussion of various methods for identifying LCSs. The utility of this more general viewpoint is highlighted through a variety of examples. We also show that although LCSs approximate GLCSs in certain limiting situations under restrictive assumptions on how the velocity field affects the additional quantities of interest, LCSs are not in general sufficient to describe their coherent transport.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2018.01.011
  • Nonequilibrium diffusive gas dynamics: Poiseuille microflow
    • Authors: Rafail V. Abramov; Jasmine T. Otto
      Abstract: Publication date: Available online 16 January 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Rafail V. Abramov, Jasmine T. Otto
      We test the recently developed hierarchy of diffusive moment closures for gas dynamics together with the near-wall viscosity scaling on the Poiseuille flow of argon and nitrogen in a one micrometer wide channel, and compare it against the corresponding Direct Simulation Monte Carlo computations. We find that the diffusive regularized Grad equations with viscosity scaling provide the most accurate approximation to the benchmark DSMC results. At the same time, the conventional Navier–Stokes equations without the near-wall viscosity scaling are found to be the least accurate among the tested closures.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2018.01.006
  • Single bumps in a 2-population homogenized neuronal network model
    • Authors: Karina Kolodina; Anna Oleynik; John Wyller
      Abstract: Publication date: Available online 11 January 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Karina Kolodina, Anna Oleynik, John Wyller
      We investigate existence and stability of single bumps in a homogenized 2-population neural field model, when the firing rate functions are given by the Heaviside function. The model is derived by means of the two-scale convergence technique of Nguetseng in the case of periodic microvariation in the connectivity functions. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. The bump solutions are constructed by using a pinning function technique for the case where the solutions are independent of the local variable. In the weakly modulated case the generic picture consists of two bumps (one narrow and one broad bump) for each admissible set of threshold values for firing. In addition, a new threshold value regime for existence of bumps is detected. Beyond the weakly modulated regime the number of bumps depends sensitively on the degree of heterogeneity. For the latter case we present a configuration consisting of three coexisting bumps. The linear stability of the bumps is studied by means of the spectral properties of a Fredholm integral operator, block diagonalization of this operator and the Fourier decomposition method. In the weakly modulated regime, one of the bumps is unstable for all relative inhibition times, while the other one is stable for small and moderate values of this parameter. The latter bump becomes unstable as the relative inhibition time exceeds a certain threshold. In the case of the three coexisting bumps detected in the regime of finite degree of heterogeneity, we have at least one stable bump (and maximum two stable bumps) for small and moderate values of the relative inhibition time.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2018.01.004
  • Traveling waves in a spatially-distributed Wilson–Cowan model of
           cortex: From fronts to pulses
    • Authors: Jeremy D. Harris; Bard Ermentrout
      Abstract: Publication date: Available online 2 January 2018
      Source:Physica D: Nonlinear Phenomena
      Author(s): Jeremy D. Harris, Bard Ermentrout
      Wave propagation in excitable media has been studied in various biological, chemical, and physical systems. Waves are among the most common evoked and spontaneous organized activity seen in cortical networks. In this paper, we study traveling fronts and pulses in a spatially-extended version of the Wilson–Cowan equations, a neural firing rate model of sensory cortex having two population types: Excitatory and inhibitory. We are primarily interested in the case when the local or space-clamped dynamics has three fixed points: (1) a stable down state; (2) a saddle point with stable manifold that acts as a threshold for firing; (3) an up state having stability that depends on the time scale of the inhibition. In the case when the up state is stable, we look for wave fronts, which transition the media from a down to up state, and when the up state is unstable, we are interested in pulses, a transient increase in firing that returns to the down state. We explore the behavior of these waves as the time and space scales of the inhibitory population vary. Some interesting findings include bistability between a traveling front and pulse, fronts that join the down state to an oscillation or spatiotemporal pattern, and pulses which go through an oscillatory instability.

      PubDate: 2018-02-05T15:49:08Z
      DOI: 10.1016/j.physd.2017.12.011
  • Inverse scattering transform and soliton solutions for square matrix
           nonlinear Schrödinger equations with non-zero boundary conditions
    • Authors: Barbara Prinari; Francesco Demontis; Sitai Li; Theodoros P. Horikis
      Abstract: Publication date: Available online 19 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Barbara Prinari, Francesco Demontis, Sitai Li, Theodoros P. Horikis
      The inverse scattering transform (IST) with non-zero boundary conditions at infinity is developed for an m × m matrix nonlinear Schrödinger-type equation which, in the case m = 2 , has been proposed as a model to describe hyperfine spin F = 1 spinor Bose–Einstein condensates with either repulsive interatomic interactions and anti-ferromagnetic spin-exchange interactions (self-defocusing case), or attractive interatomic interactions and ferromagnetic spin-exchange interactions (self-focusing case). The IST for this system was first presented by Ieda, Uchiyama and Wadati (2007) , using a different approach. In our formulation, both the direct and the inverse problems are posed in terms of a suitable uniformization variable which allows to develop the IST on the standard complex plane, instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity of the scattering eigenfunctions and scattering data, symmetries, properties of the discrete spectrum, and asymptotics are derived. The inverse problem is posed as a Riemann-Hilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided. In addition, the general behavior of the soliton solutions is analyzed in details in the 2  ×  2 self-focusing case, including some special solutions not previously discussed in the literature.

      PubDate: 2017-12-27T13:08:29Z
      DOI: 10.1016/j.physd.2017.12.007
  • Poisson-Nernst-Planck equations with steric effects - non-convexity and
           multiple stationary solutions
    • Authors: Nir Gavish
      Abstract: Publication date: Available online 16 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Nir Gavish
      We study the existence and stability of stationary solutions of Poisson-Nernst-Planck equations with steric effects (PNP-steric equations) with two counter-charged species. We show that within a range of parameters, steric effects give rise to multiple solutions of the corresponding stationary equation that are smooth. The PNP-steric equation, however, is found to be ill-posed at the parameter regime where multiple solutions arise. Following these findings, we introduce a novel PNP-Cahn-Hilliard model, show that it is well-posed and that it admits multiple stationary solutions that are smooth and stable. The various branches of stationary solutions and their stability are mapped utilizing bifurcation analysis and numerical continuation methods.

      PubDate: 2017-12-18T12:46:36Z
      DOI: 10.1016/j.physd.2017.12.008
  • Computing Evans functions numerically via boundary-value problems
    • Authors: Blake Barker; Rose Nguyen; Björn Sandstede; Nathaniel Ventura; Colin Wahl
      Abstract: Publication date: Available online 9 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Blake Barker, Rose Nguyen, Björn Sandstede, Nathaniel Ventura, Colin Wahl
      The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this paper, an alternative scheme for the numerical computation of Evans functions is presented that relies on an appropriate boundary-value problem formulation. Convergence of the algorithm is proved, and several examples, including the computation of eigenvalues for a multi-dimensional problem, are given. The main advantage of the scheme proposed here compared with earlier methods is that the scheme is linear and scalable to large problems.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.12.002
  • Turing patterns in parabolic systems of conservation laws and numerically
           observed stability of periodic waves
    • Authors: Blake Barker; Soyeun Jung; Kevin Zumbrun
      Abstract: Publication date: Available online 7 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Blake Barker, Soyeun Jung, Kevin Zumbrun
      Turing patterns on unbounded domains have been widely studied in systems of reaction–diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from subcritical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram–specifically, determination of rigorous Eckhaus-type stability conditions–remains an interesting open problem.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.12.003
  • Travelling waves and their bifurcations in the Lorenz-96 model
    • Authors: Dirk L. van Kekem; Alef E. Sterk
      Abstract: Publication date: Available online 5 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Dirk L. van Kekem, Alef E. Sterk
      In this paper we study the dynamics of the monoscale Lorenz-96 model using both analytical and numerical means. The bifurcations for positive forcing parameter F are investigated. The main analytical result is the existence of Hopf or Hopf-Hopf bifurcations in any dimension n ≥ 4 . Exploiting the circulant structure of the Jacobian matrix enables us to reduce the first Lyapunov coefficient to an explicit formula from which it can be determined when the Hopf bifurcation is sub- or supercritical. The first Hopf bifurcation for F > 0 is always supercritical and the periodic orbit born at this bifurcation has the physical interpretation of a travelling wave. Furthermore, by unfolding the codimension two Hopf-Hopf bifurcation it is shown to act as an organising centre, explaining dynamics such as quasi-periodic attractors and multistability, which are observed in the original Lorenz-96 model. Finally, the region of parameter values beyond the first Hopf bifurcation value is investigated numerically and routes to chaos are described using bifurcation diagrams and Lyapunov exponents. The observed routes to chaos are various but without clear pattern as n → ∞ .

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.008
  • New integrable model of propagation of the few-cycle pulses in an
           anisotropic microdispersed medium
    • Authors: S.V. Sazonov; N.V. Ustinov
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): S.V. Sazonov, N.V. Ustinov
      We investigate the propagation of the few-cycle electromagnetic pulses in the anisotropic microdispersed medium. The effects of the anisotropy and spatial dispersion of the medium are created by the two sorts of the two-level atoms. The system of the material equations describing an evolution of the states of the atoms and the wave equations for the ordinary and extraordinary components of the pulses is derived. By applying the approximation of the sudden excitation to exclude the material variables, we reduce this system to the single nonlinear wave equation that generalizes the modified sine–Gordon equation and the Rabelo–Fokas equation. It is shown that this equation is integrable by means of the inverse scattering transformation method if an additional restriction on the parameters is imposed. The multisoliton solutions of this integrable generalization are constructed and investigated.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.012
  • Decay of Kadomtsev–Petviashvili lumps in dissipative media
    • Authors: S. Clarke; K. Gorshkov; R. Grimshaw; Y. Stepanyants
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): S. Clarke, K. Gorshkov, R. Grimshaw, Y. Stepanyants
      The decay of Kadomtsev–Petviashvili lumps is considered for a few typical dissipations –Rayleigh dissipation, Reynolds dissipation, Landau damping, Chezy bottom friction, viscous dissipation in the laminar boundary layer, and radiative losses caused by large-scale dispersion. It is shown that the straight-line motion of lumps is unstable under the influence of dissipation. The lump trajectories are calculated for two most typical models of dissipation –the Rayleigh and Reynolds dissipations. A comparison of analytical results obtained within the framework of asymptotic theory with the direct numerical calculations of the Kadomtsev–Petviashvili equation is presented. Good agreement between the theoretical and numerical results is obtained.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.009
  • Interaction of non-radially symmetric camphor particles
    • Authors: Shin-Ichiro Ei; Hiroyuki Kitahata; Yuki Koyano; Masaharu Nagayama
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Shin-Ichiro Ei, Hiroyuki Kitahata, Yuki Koyano, Masaharu Nagayama
      In this study, the interaction between two non-radially symmetric camphor particles is theoretically investigated and the equation describing the motion is derived as an ordinary differential system for the locations and the rotations. In particular, slightly modified non-radially symmetric cases from radial symmetry are extensively investigated and explicit motions are obtained. For example, it is theoretically shown that elliptically deformed camphor particles interact so as to be parallel with major axes. Such predicted motions are also checked by real experiments and numerical simulations.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.004
  • Modeling ultrashort electromagnetic pulses with a generalized
           Kadomtsev–Petviashvili equation
    • Authors: A. Hofstrand; J.V. Moloney
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): A. Hofstrand, J.V. Moloney
      In this paper we derive a properly scaled model for the nonlinear propagation of intense, ultrashort, mid-infrared electromagnetic pulses (10-100 femtoseconds) through an arbitrary dispersive medium. The derivation results in a generalized Kadomtsev–Petviashvili (gKP) equation. In contrast to envelope-based models such as the Nonlinear Schrödinger (NLS) equation, the gKP equation describes the dynamics of the field’s actual carrier wave. It is important to resolve these dynamics when modeling ultrashort pulses. We proceed by giving an orginal proof of sufficient conditions on the initial pulse for a singularity to form in the field after a finite propagation distance. The model is then numerically simulated in 2D using a spectral-solver with initial data and physical parameters highlighting our theoretical results.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.010
  • Global dynamics for switching systems and their extensions by linear
           differential equations
    • Authors: Zane Huttinga; Bree Cummins; Tomáš Gedeon; Konstantin Mischaikow
      Abstract: Publication date: Available online 15 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Zane Huttinga, Bree Cummins, Tomáš Gedeon, Konstantin Mischaikow
      Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.003
  • Three-wave resonant interactions: Multi-dark-dark-dark solitons,
           breathers, rogue waves, and their interactions and dynamics
    • Authors: Guoqiang Zhang; Zhenya Yan; Xiao-Yong Wen
      Abstract: Publication date: Available online 10 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Guoqiang Zhang, Zhenya Yan, Xiao-Yong Wen
      We investigate three-wave resonant interactions through both the generalized Darboux transformation method and numerical simulations. Firstly, we derive a simple multi-dark-dark-dark-soliton formula through the generalized Darboux transformation. Secondly, we use the matrix analysis method to avoid the singularity of transformed potential functions and to find the general nonsingular breather solutions. Moreover, through a limit process, we deduce the general rogue wave solutions and give a classification by their dynamics including bright, dark, four-petals, and two-peaks rogue waves. Ever since the coexistence of dark soliton and rogue wave in non-zero background, their interactions naturally become a quite appealing topic. Based on the N -fold Darboux transformation, we can derive the explicit solutions to depict their interactions. Finally, by performing extensive numerical simulations we can predict whether these dark solitons and rogue waves are stable enough to propagate. These results can be available for several physical subjects such as fluid dynamics, nonlinear optics, solid state physics, and plasma physics.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.001
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